Theorem isso2i 5067

Description: Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
isso2i.1	|- ( ( x e. A /\ y e. A ) -> ( x R y <-> -. ( x = y \/ y R x ) ) )
isso2i.2	|- ( ( x e. A /\ y e. A /\ z e. A ) -> ( ( x R y /\ y R z ) -> x R z ) )

Assertion

Ref	Expression
isso2i	|- R Or A

Distinct variable groups:   x, y, z, R   x, A, y, z

Proof of Theorem isso2i

Step	Hyp	Ref	Expression
1		equid 1939	. . . . 5 |- x = x
2	1	orci 405	. . . 4 |- ( x = x \/ x R x )
3		eleq1 2689	. . . . . . 7 |- ( y = x -> ( y e. A <-> x e. A ) )
4	3	anbi2d 740	. . . . . 6 |- ( y = x -> ( ( x e. A /\ y e. A ) <-> ( x e. A /\ x e. A ) ) )
5		equequ2 1953	. . . . . . . 8 |- ( y = x -> ( x = y <-> x = x ) )
6		breq1 4656	. . . . . . . 8 |- ( y = x -> ( y R x <-> x R x ) )
7	5, 6	orbi12d 746	. . . . . . 7 |- ( y = x -> ( ( x = y \/ y R x ) <-> ( x = x \/ x R x ) ) )
8		breq2 4657	. . . . . . . 8 |- ( y = x -> ( x R y <-> x R x ) )
9	8	notbid 308	. . . . . . 7 |- ( y = x -> ( -. x R y <-> -. x R x ) )
10	7, 9	bibi12d 335	. . . . . 6 |- ( y = x -> ( ( ( x = y \/ y R x ) <-> -. x R y ) <-> ( ( x = x \/ x R x ) <-> -. x R x ) ) )
11	4, 10	imbi12d 334	. . . . 5 |- ( y = x -> ( ( ( x e. A /\ y e. A ) -> ( ( x = y \/ y R x ) <-> -. x R y ) ) <-> ( ( x e. A /\ x e. A ) -> ( ( x = x \/ x R x ) <-> -. x R x ) ) ) )
12		isso2i.1	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> ( x R y <-> -. ( x = y \/ y R x ) ) )
13	12	con2bid 344	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( ( x = y \/ y R x ) <-> -. x R y ) )
14	11, 13	chvarv 2263	. . . 4 |- ( ( x e. A /\ x e. A ) -> ( ( x = x \/ x R x ) <-> -. x R x ) )
15	2, 14	mpbii 223	. . 3 |- ( ( x e. A /\ x e. A ) -> -. x R x )
16	15	anidms 677	. 2 |- ( x e. A -> -. x R x )
17		isso2i.2	. 2 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( ( x R y /\ y R z ) -> x R z ) )
18	13	biimprd 238	. . . 4 |- ( ( x e. A /\ y e. A ) -> ( -. x R y -> ( x = y \/ y R x ) ) )
19	18	orrd 393	. . 3 |- ( ( x e. A /\ y e. A ) -> ( x R y \/ ( x = y \/ y R x ) ) )
20		3orass 1040	. . 3 |- ( ( x R y \/ x = y \/ y R x ) <-> ( x R y \/ ( x = y \/ y R x ) ) )
21	19, 20	sylibr 224	. 2 |- ( ( x e. A /\ y e. A ) -> ( x R y \/ x = y \/ y R x ) )
22	16, 17, 21	issoi 5066	1 |- R Or A

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   e. wcel 1990   class class class wbr 4653   Or wor 5034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-po 5035  df-so 5036

This theorem is referenced by:  ltsonq  9791  ltsosr  9915  ltso  10118  xrltso  11974